Northern lights effect along line on textured plane
You are probably better off creating an actual curve for the path of your effect than doing it from a "texture". Having actual geometry will help.

Just draw a bezier curve in Blender following the desired path, or draw it in Photoshop or Illustrator or Inkscape if you prefer, just make it so it is an actual Bezier Vector Curve, and can be exported as an SVG file and then imported into Blender.

Extrude said curve and then add the material.

The material is basically just a gradient in Z direction, from an Emission to an Transparent shader. Use the same Gradient texture, with Generated texture coordinates (so it adapts automatically to size changes) to drive both the emission progression of the Aurora Borialis and the color variation, as shown bellow.